Permutation theory governs long-term dynamics of critical Boolean networks

Abstract

Boolean networks are widely used to model gene regulatory Attractors of Boolean networks model stable gene-expression patterns, yet deriving their properties from network structure remains an open problem. We solve this problem for critical $K=1$ networks by showing that their feedback loops induce a permutation whose order bounds the average attractor length both above and below by universal constant factors. This correspondence allows classical results from combinatorics and number theory to be applied directly to Boolean network dynamics. We find three distinct asymptotic scales for both average and maximum attractor lengths: typical networks scale as $\exp[(1/8+o(1))(\ln N)^2]$, the ensemble means grow as $\exp[N^{1/3+o(1)}]$, and extremal networks attain $\exp[(1+o(1))\sqrt{N\ln N}]$. Thus, ensemble averages are governed by rare network realizations and are unrepresentative of typical dynamics.